Answer:
[tex]3x^2-5x=-8[/tex]
[tex]2x^2=6x-5[/tex]
[tex]-x^2-10x=34[/tex]
Step-by-step explanation:
The equations are:
[tex]3x^2-5x=-8[/tex]
[tex]2x^2=6x-5[/tex]
[tex]12x=9x^2+4[/tex]
[tex]-x^2-10x=34[/tex]
(a) [tex]3x^2-5x=-8[/tex]
Add 8 to both sides
[tex]3x^2 - 5x + 8 =0[/tex]
To do this, we simply calculate the discriminant (d) using:
[tex]d =b^2 - 4ac[/tex]
If [tex]d < 0[/tex]
Then it has complex roots
Where: [tex]a = 3; b = -5; c = 8[/tex]
[tex]d = (-5)^2 - 4 * 3 * 8[/tex]
[tex]d = 25 - 96[/tex]
[tex]d = -71[/tex]
[tex]-71 < 0[/tex] --- complex root
(b) [tex]2x^2=6x-5[/tex]
Equate to 0
[tex]2x^2 - 6x + 5 = 0[/tex]
[tex]d =b^2 - 4ac[/tex]
[tex]d = (-6)^2 - 4 * 2 * 5[/tex]
[tex]d = 36 - 40[/tex]
[tex]d =-4[/tex]
[tex]-4 < 0[/tex] --- complex root
(c) [tex]12x=9x^2+4[/tex]
Equate to 0
[tex]9x^2 - 12x + 4 = 0[/tex]
[tex]d =b^2 - 4ac[/tex]
[tex]d = (-12)^2 - 4 * 9 * 4[/tex]
[tex]d = 144 - 144[/tex]
[tex]d = 0[/tex] ---- real roots
(d) [tex]-x^2-10x=34[/tex]
Equate to 0
[tex]-x^2 - 10x - 34 = 0[/tex]
[tex]d =b^2 - 4ac[/tex]
[tex]d = (-10)^2 - 4 * (-1) * (-34)[/tex]
[tex]d = 100 - 136[/tex]
[tex]d = - 36[/tex]
[tex]-36 < 0[/tex] ---- complex root
